Find y as a function of t if 324y′′+684y′+361y=0, y(0)=9,y′(0)=9.

Q: Compute the values of dy and Ay for the function y= e4z + 5z given z = 0 and Az = dz = 0.02.

A: Given x = 0, dx = ∆x = 0.02            (1) Here y = f(x) = e4x+5x

Q: find the directions in which the functions increase and decrease most rapidly at P0 . Then find the…

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Q: 4. For the below function, find the stationary point (s) and prove if a maximum, minimum, or saddle…

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Q: Let z = f(x, y) = 5x – x³ + 2xy – y². - Find and classify all the critical point(s) of this…

A: Differentiate the function fx,y=5x-x3+2xy-y2 partially with respect to x and y, to obtain fx and fy.…

Q: Find the linearization L(x,y) of the function f(x,y) = x2 - 4xy + 5 at Po(5,4).

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Q: 2²y. 3. Consider the function f(, y) =

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Q: The function g(t) = \t| is differentiable at 0. True False

A: Given that g(t)=|t|We can write it as:g(t)=t   , t≥0-t , t<0

Q: 4. Find if y is a function of r defined implicitly by da 2*y + sin a = xe" + In

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Q: Differentiate the function. y = In(16 + t – P) %3D y' = 1– 3r² 372 6 +t-t -- 6 + t- r 2 6+t-t

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Q: Find y as a function of t if y" + y' – 2y = 0, y(0) = 8, y(1) = 8. y(t) = Remark: The initial…

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Q: Find y as a function of t if y(0) = 4, y(0) = 1. y(t): = 64y" - 144y + 81y = 0,

A: Given: The characteristics equation is             64m2-144m+81=0⇒m=--144±-1442-4· 64· 812·…

Q: 3. Find the linearization L(x) of the function at a. F(x) = Vx, a = 4

A: Use the formula for the linearization of a function about any point.

Q: of and dy For the function f(x,y) = In (8x + 9y), find - - of

A: partial derivative of two variable functions

Q: Find fryz from the functions f (x, y, z) = 4x³y² – e²y+ ++ 4y – x16

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Q: Find y as a function of t if y" – 10y' + 21y = 0, y(0) = 7, y(1) = 4. y(t) = Remark: The initial…

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Q: For the given function 9 = x-yz and 62 -(@C) at C= 3x?yi+ yz²j- xzk, Find 5yöz бубz point (1, – 2, –…

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Q: 2. Show that the function u p(xy) + Vry (y/x) provides the equation = 0.

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Q: Find the ditterential of the function y = (x + 1(5x - 3). Sx-x-5 dy = (Sx - 3) *?+ 6x-5 (5x- 3)? dy…

A: Use the quotient rule of differentiation ddxuv=vdudx-udvdxv2. Power rule of differentiation…

Q: For the function f(x, y) = (cos x)ey +v 1 + x + y. %3D Show the process of using the linearization…

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Q: YouTrylt 2 Find the linear approximation of the function f(x,y) =In(x- 3y) (7,2) at and use it to…

A: Please rate and feel free to ask any query about any part of the question.  Thanks

Q: ) Find the local linear approximation of the function f(x) = V1+x at xo = 3, and use it to…

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Q: Find an equation of the tangent line to the graph of f (t) = (t – 5) (t² – 3) at the given point (2,…

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Q: Find x' for x(t) defined implicitly by x+t°x+t -8=0 and then evaluate x' at the point (-2,2).

A: x3+t3x+t4-8=0Differentiate w.r.t…

Q: Find x' for x(t) defined implicitly by x +tx+t° - 5=0 and then evaluate x' at the point (- 1, - 3).

A: Follow the procedure given below.

Q: Find of the function f(x,y) = sin(x²y³). əx3

A: The product rule of differentiation : (PQ)'=PQ'+QP' sinx'=cosxcosx'=-sinxxn'=nxn-1 Use the given…

Q: Find the vertical trace of the function at x = 1: g(x, y) = cos√x² + y²

A: Definition : Consider a function z=f(x,y) with domain D⊆R2. A vertical trace of the function can be…

Q: Find the linearization f(x) = √x, L(x) = L(x) of the function at a. a=4

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Q: Verify that the indicated function y = p(x) is an explicit solution of the given first-order…

A: This question can be solved using the concept of derivatives.

Q: Find the linear approximation of the function at the given point. f(x, y) = -4x²y³ at (-9,7) O A.…

A: In this question we have to find linear approximation of the given function.

Q: Find fryz from the functions f (x, y, z) = 4x³y² – e²y* + + 4y – x16

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Q: Find the differential of the function. z = 7t − 7/t2

A: zt=7t-7t2

Q: Find the linearization of the function f(x): L(z)= ab a --4x-5 va a न 4x+3 at z=-1 sin (a) E

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Q: Find the critical point(s) of the function f(x, y) = x² – 3xy+5x – 6y +8, and determine whether they…

A: Given function is  fx,y=x2-3xy+5x-6y+8 The function f is defined for all x,y∈ℝ2 and partial…

Q: Find the linearization L(æ) of the function at a. f(z) = V7, a = -8 L(x)

A: We can find the linearization as below.

Q: By direct differentiation of x(t) = eAtxo + eA(t-o) Bu(o)do show that ¿(t) = Ax(t) + Bu(t), x(0) =…

A: The differentiation of a definite integral can be obtained by using the Chain Rule following…

Q: Q.8 Find the gradient Vw ofthe function w = In /x² + y² + z². %3D

A: Given function W=ln(√(x2+y2+z2)) Now we will find the gradient as shown below

Q: Let z = g(x, y) be a function defined by the implicit equation xe2z+2 – 5/x² + y² = 10x – 22y + 36.…

A: Given equation z=g(x,y)  is  xe2z+2-5x2+y2=10x-22y+36  To determine a equation of tangent plane at…

Q: Find the linear approximation 0 to Write your answer in the form y = Ax + B. at x =

A: Explanation: The Linear Approximation should be a line that can substitute (in a narrow interval)…

Q: Find any value(s) of the constant a such that the line y = ax is tangent to the curve f(x) = -x² +…

A: Tangent of a curved touches the curve only at one point.

Q: Find an equation for the level surface of the function f (x, y) = ln (x² + y²+ z²) that passes…

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Q: A function y(t) satisfies th

A: Given, the differential equation                       dydt=-y4-3y3+4y2

Q: 5 B) Find y" for the function y = x + Vx at p(1,3)

A: Given query is to find the double derivative of the expression at point p.

Q: 2. Show that the function u - p(sy) + Vj (y/2) provides the equation 0.

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Q: for the function f(Z) z²-2z find the (z+1)° (2 t4) residues at singular points.

A: The residue of f(z) is solved below

Q: 1. Find the stationary points of the following functions. (a) f(x,y) = x²y+ 2y² – 2xy + 6 - (b)…

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Q: Q3: Find fxx of the function f (x, y, z) = -In/1-x2 - y2 – z2 %3D

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Q: Find the linearization of the function at the given point f(x,y)=-10x+8y-9 at (-7,4). A.…

A: I have appended the formula used in the solution part

Q: Find the G.S. about xo = 0 for: (x² + 4)y" + xy = x + 2

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Q: Find y as a function of t if y(t) y" + 2y + 10y = 0, y(0) = 8, y'(0) = 9.

A: The given DE is: y''+2y'+10y=0 The auxiliary equation of the given DE is:…

Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differential equations.